On the Smallest \((1, 2)\)-Eulerian Weight of Graphs

A weight \(w: E(G) \rightarrow \{1, 2\}\) is called a \((1, 2)\)-eulerian weight of graph \(G\) if the total weight of each edge-cut is even. A \((1, 2)\)-eulerian weight \(w\) of \(G\) is called smallest if the total weight \(w\) of \(G\) is minimum. In this note, we prove that if graph \(G\) is \(2\)-connected and simple, and \(w_0\) is a smallest \((1, 2)\)-eulerian weight, then either \(|E_{w_0 = \text{even}}|\leq|V(G)| – 3\) or \(G = K_4\).